Question

If \( (1,3) \) is the midpoint of a chord of the circle \( x^2 + y^2 - 4x - 8y + 16 = 0 \), then the area of the triangle formed by that chord with the coordinate axes is:

• A. \( 16 \)
• B. \( 8 \)
• C. \( 4 \)
• D. \( 8\sqrt{2} \)

Hint:

For a midpoint chord problem, use the midpoint theorem for chords and find the intercepts to compute the area.

Answer 1

The Correct Option is B

Step 1: Rewrite the circle equation in standard form The given circle equation is: \[ x^2 + y^2
- 4x
- 8y + 16 = 0 \] Complete the square for \( x \) and \( y \): \[ x^2
- 4x + y^2
- 8y =
-16 \] For \( x \): \[ x^2
- 4x = (x
- 2)^2
- 4 \] For \( y \): \[ y^2
- 8y = (y
- 4)^2
- 16 \] Substitute back into the equation: \[ (x
- 2)^2
- 4 + (y
- 4)^2
- 16 =
-16 \] Simplify: \[ (x
- 2)^2 + (y
- 4)^2
- 20 =
-16
(x
- 2)^2 + (y
- 4)^2 = 4 \] This is the standard form of a circle with center \( (2, 4) \) and radius \( 2 \). Step 2: Find the equation of the chord The chord has \( (1, 3) \) as its midpoint. The line from the center of the circle \( (2, 4) \) to the midpoint \( (1, 3) \) is perpendicular to the chord. The slope of this line is: \[ m_1 = \frac{3
- 4}{1
- 2} = \frac{
-1}{
-1} = 1 \] Since the chord is perpendicular to this line, its slope \( m_2 \) satisfies \( m_1 \cdot m_2 =
-1 \). Thus: \[ m_2 =
-1 \] The equation of the chord, using the point
-slope form and the midpoint \( (1, 3) \), is: \[ y
- 3 =
-1(x
- 1)
y
- 3 =
-x + 1
x + y
- 4 = 0 \] Step 3: Find the points where the chord intersects the axes The chord intersects the \( x \)
-axis when \( y = 0 \). Substitute \( y = 0 \) into \( x + y
- 4 = 0 \): \[ x + 0
- 4 = 0
x = 4 \] The chord intersects the \( y \)
-axis when \( x = 0 \). Substitute \( x = 0 \) into \( x + y
- 4 = 0 \): \[ 0 + y
- 4 = 0
y = 4 \] Thus, the chord intersects the axes at \( (4, 0) \) and \( (0, 4) \). Step 4: Compute the area of the triangle The triangle is formed by the points \( (4, 0) \), \( (0, 4) \), and the origin \( (0, 0) \). The area \( A \) of the triangle is: \[ A = \frac{1}{2} \cdot \text{base} \cdot \text{height} \] Here, the base is \( 4 \) (distance along the \( x \)
-axis) and the height is \( 4 \) (distance along the \( y \)
-axis). Thus: \[ A = \frac{1}{2} \cdot 4 \cdot 4 = 8 \] Final Answer: The area of the triangle is \( 8 \). \[ \boxed{8} \]